Multinomial logistic regression models are popular in multicategory classification analysis, but existing models suffer several intrinsic drawbacks. In particular, the parameters cannot be determined uniquely because of the over-specification. Although additional constraints have been imposed to refine the model, such modifications can be inefficient and complicated. In this paper, we propose a novel and efficient simplex-based multinomial logistic regression technique, seamlessly connecting binomial and multinomial cases under a unified framework. Compared with existing models, our model has fewer parameters, is free of any constraints, and can be solved efficiently using the Fisher scoring algorithm. In addition, the proposed model enjoys several theoretical advantages, including Fisher consistency and sharp comparison inequality. Under mild conditions, we establish the asymptotical normality and convergence for the new model, even when the numbers of categories and covariates increase with the sample size. The proposed framework is illustrated by means of extensive simulations and real applications.

@en

Problem definition: Warranty reserves are funds used to fulfill future warranty obligations for a product. In this paper, we investigate the warranty reserve planning problem faced by a manufacturing firm who manages warranties for multiple products. Academic/practical relevance: It is nontrivial to determine a proper amount of reserves to hold, because warranty expenditures are random in nature and reserving either excess or insufficient cash would incur losses. How can warranty reserve levels be optimized and promptly adjusted is a focal issue, especially for firms selling multiple products. Methodology: Inspired by the general pattern of empirical warranty claims data, we first develop an aggregate warranty cost (AWC) forecasting model for a single product by coupling stochastic product sales and failure processes, which can be used to plan for warranty reserves periodically. The reserve levels are then optimized via a distributionally robust approach, because the exact distribution of AWC is generally unknown. To reduce the losses generated from managing the funds, we further investigate two potential loss-reduction approaches: demand learning and funds pooling. Results: For the demand learning algorithm, we prove that, as the sales period grows, the optimal learning parameter asymptotically converges to a constant in a fairly fast rate; our simulation experiments show that the performance of demand learning is promising and robust under general warranty claim patterns. Moreover, we find that the benefits of funds pooling change over different stages of the warranty life cycle; in particular, the relative pooling benefit in terms of reserve losses is nonincreasing over time. Managerial implications: This study offers guidelines on how manufacturers should adaptively forecast and dynamically plan warranty reserves over the warranty life cycle.

@en

This study proposes an efficient approach to detect one or more change points for gamma distribution. We plug a closed-form estimator into the gamma log-likelihood function to obtain a sharp approximation to the maximum of log-likelihood. We further derive a closed form calibration of approximate likelihood which is asymptotically equivalent to the exact log-likelihood. This circumvents iterative optimization procedures to find maximum likelihood estimates which can be a burden in detecting multiple change points. The simulation study shows that the approximation is accurate and the change points can be detected much faster. Two case studies on the time between events arising from industrial accidents are presented and extensively investigated.

@en

Many large organizations have developed ambitious programs to build reliability databases by collecting field failure data from a large variety of components. To make the database concise, only the number of component replacements in a component position during an operation time interval is reported in these databases. This leads to time-censoring in the aggregate failure data. Statistical inference for the time-censored aggregate data is challenging, because the likelihood function based on some common lifetime distributions can be intractable. In this study, we propose a general parametric estimation framework for the aggregate data. We first use the gamma distribution and the Inverse Gaussian (IG) distribution to model the aggregate data. Bayesian inference for the two models is discussed. Unlike the gamma/IG distribution, other lifetime distributions involve multiple integrals in the likelihood function, making the standard Bayesian inference difficult. To address the estimation problem, an approximate Bayesian computation algorithm that does not require evaluating the likelihood function is proposed, and its performance is assessed by simulation. As there are several candidate distributions, we further propose a model selection procedure to identify an appropriate distribution for the time-censored aggregate data. A real aggregate dataset extracted from a reliability database is used for illustration.

@en

Process capability indices (PCIs) are widely used to assess whether an in-control process meets manufacturing specifications. In most applications of classical PCIs, the process characteristic is assumed normally distributed. However, the normal distribution has been found inappropriate in various applications. In the literature, the percentile-based PCIs are widely used to deal with the nonnormal process. One problem associated with the percentile-based PCIs is that they do not provide a quantitative interpretation to the process capability. In this study, new PCIs that have a consistent quantification to the process capability for both normal and nonnormal processes are proposed. The proposed PCIs are generalizations of the classical normal PCIs in the sense that they are the same as the classical PCIs when the process characteristic follows a normal distribution, and they offer the same interpretation to the process capability as the classical PCIs when the process characteristic is nonnormal. We then discuss nonparametric and parametric estimation of the proposed PCIs. The nonparametric estimator is based on the kernel density estimation and confidence limits are obtained by the nonparametric bootstrap, while the parametric estimator is based on the maximum likelihood estimation and confidence limits are constructed by the method of generalized pivots. The proposed methodologies are demonstrated using a real example from a manufacturing factory.@en

Reliability data obtained from life tests and degradation tests have been extensively used for purposes such as estimating product reliability and predicting warranty costs. When there is more than one candidate model, an important task is to discriminate between the models. In the literature, the model discrimination was often treated as a hypothesis test and a pairwise model discrimination procedure was carried out. Because the null distribution of the test statistic is unavailable in most cases, the large sample approximation and the bootstrap were frequently used to find the acceptance region of the test. Although these two methods are asymptotically accurate, their performance in terms of size and power is not satisfactory in small sample size. To enhance the small-sample performance, we propose a new method to approximate the null distribution, which builds on the idea of generalized pivots. Conventionally, the generalized pivots were often used for interval estimation of a certain parameter or function of parameters in presence of nuisance parameters. In this study, we further extend the idea of generalized pivots to find the acceptance region of the model discrimination test. Through extensive simulations, we show that the proposed method performs better than the existing methods in discriminating between two lifetime distributions or two degradation models over a wide range of sample sizes. Two real examples are used to illustrate the proposed methods.

@en

Process capability indices (PCIs) are widely used to measure whether an in-control process conforms to manufacturing specifications. The normal distribution is assumed in most traditional applications of PCIs. Nevertheless, it is not uncommon that some quality characteristics have skewed distributions. In such cases, the gamma distribution is an appropriate model and percentile-based PCIs for the gamma process have been studied in the literature. In practical applications of PCIs, it is important to select an appropriate distribution between the normal and the gamma distributions based on historical data. In this study, we first construct a hypothesis test for model discrimination between the normal and the gamma distributions. Asymptotic distribution of the test statistic under the gamma process is derived. We then consider statistical inference for the percentile-based PCIs under the gamma process. The maximum likelihood method is used for point estimation and the method of generalized pivotal quantities is used for interval estimation. We demonstrate the proposed methods by a practical example.@en

Degradation data are an important source of product reliability information. Two popular stochastic models for degradation data are the Gamma process and the inverse Gaussian (IG) process, both of which possess monotone degradation paths. Although these two models have been used in numerous applications, the existing interval estimation methods are either inaccurate given a moderate sample size of the degradation data or require a significant computation time when the size of the degradation data is large. To bridge this gap, this article develops a general framework of interval estimation for the Gamma and IG processes based on the method of generalized pivotal quantities. Extensive simulations are conducted to compare the proposed methods with existing methods under moderate and large sample sizes. Degradation data from capacitors are used to illustrate the proposed methods.

@en

Field data provide important information about product quality and reliability. Many large organizations have developed ambitious reliability databases to trace field failure data of a variety of components on the systems they operate and maintain. Due to the exponential distribution assumption for the component lifetimes, the data in these databases are often aggregated. Specifically, individual lifetimes of the components are not available. Instead, each recorded data point is the cumulative operating time of one component position from system installation to the last component replacement, and the number of replacements in between. In the literature, the gamma distribution and the inverse Gaussian (IG) distribution have been used to fit the aggregate data, while the operating environment of different systems is often assumed the same. In order to capture possible heterogeneities among the systems, this study proposes the gamma random effects model and the IG random effects model. The expectation-maximization algorithm is used for point estimation of the parameters and an algorithm based on the generalized fiducial inference method is proposed for interval estimation. Simulation studies are conducted to assess the performance of the proposed inference methods. A real aggregate dataset is used for illustration.

@en

The generalized Pareto distribution (GPD) is widely used for extreme values over a threshold. Most existing methods for parameter estimation either perform unsatisfactorily when the shape parameter k is larger than 0.5, or they suffer from heavy computation as the sample size increases. In view of the fact that k > 0.5 is occasionally seen in numerous applications, including two illustrative examples used in this study, we remedy the deficiencies of existing methods by proposing two new estimators for the GPD parameters. The new estimators are inspired by the minimum distance estimation and the M-estimation in the linear regression. Through comprehensive simulation, the estimators are shown to perform well for all values of k under small and moderate sample sizes. They are comparable to the existing methods for k < 0.5 while perform much better for k > 0.5.

@en

Because of the exponential distribution assumption, many reliability databases recorded data in an aggregate way. Instead of individual failure times, each aggregate data point is a summation of a series of collective failures representing the cumulative operating time of one component position from system commencement to the last component replacement. The data format is different from traditional lifetime data and the statistical inference is challenging. We first model the individual component lifetime by a gamma distribution. Confidence intervals for the gamma shape parameter can be constructed using a scaled χ² approximation to a modified ratio of the geometric mean to the arithmetic mean, while confidence intervals for the gamma rate and mean parameters, as well as quantiles, are obtained using the generalized pivotal quantity method. We then fit the data using the inverse Gaussian (IG) distribution, a useful lifetime model for failures caused by degradation. Procedures for point estimation and interval estimation of parameters are developed. We also propose an interval estimation method for the quantiles of an IG distribution based on the generalized pivotal quantity method. An illustrative example demonstrates the proposed inference methods. Supplementary materials for this article are available online.

@en

This study develops methods for constructing some important statistical limits of a gamma distribution. First, we construct upper prediction limits and tolerance limits for a gamma distribution. In addition, upper prediction limits for at least p of m measurements from a gamma distribution at each of r locations are constructed. This problem often arises in the field of environmental monitoring, as well as quality control. For each problem discussed in this study, the inferential procedure based on the generalized fiducial method is outlined and a simulation is conducted to assess its performance. Real data are used to demonstrate the proposed methods. In addition to the prediction and tolerance limits, our proposed methods can also be applied to the stress-strength problem involving two independent gamma random variables. Our investigation shows that the proposed methods perform uniformly well and they outperform existing methods.

@en

The field failure data is important in the reliability field. Component failure data, instead of system failure data, are usually collected by the system owners. However, in many cases the individual component time-to-failure data are not recorded. Instead, cumulative operation time and the number of failure times are recorded. This can be explained by the fact that exponential distribution is often used to fit the data. Unfortunately, the assumption of exponential distribution is not valid in many cases. In our paper, Gamma distribution and Inverse Gaussian (IG) distribution are used to fit the data. The point estimate for parameters of these two distributions can be obtained easily by general computer software. We then proposed methods to obtain the confidence interval (CI) for parameters of gamma distribution and IG distribution when the data are merged. In particular, for the rate parameter and mean of a gamma distribution, no efficient methods have been found to get the interval estimation. We borrow the idea of General Pivotal Quantity to obtain the interval estimation for the rate parameter and mean of a gamma distribution. The simulation study shows our method outperforms Wald method in terms of coverage probability. At last, an illustrative example is given to demonstrate the applicability of proposed models.@en

In addition to conventional censoring schemes such as Type-I or Type-II censoring, progressively censoring is a useful method to reduce cost and obtain additional reliability information in accelerated life testing. Statistical inference for accelerated life testing (ALT) with Weibull distribution and progressively censoring is found to be difficult. This paper develops generalized fiducial inference techniques for the constant-stress ALT model with Weibull distribution and progressively Type-II censoring. Simulation studies reveal the good performance of the proposed inference methods. An example is given for illustration.

@en